Studying (non-planar) road networks through an algorithmic lens

This paper studies real-world road networks from an algorithmic perspective, focusing on empirical studies that yield useful properties of road networks that can be exploited in the design of fast algorithms that deal with geographic data. Unlike previous approaches, our study is not based on the assumption that road networks are planar graphs. Indeed, based on the a number of experiments we have performed on the road networks of the 50 United States and District of Columbia, we provide strong empirical evidence that road networks are quite non-planar. Our approach therefore instead is directed at finding algorithmically-motivated properties of road networks as non-planar geometric graphs, focusing on alternative properties of road networks that can still lead to efficient algorithms for such problems as shortest paths and Voronoi diagrams. In particular, we study road networks as multiscale-dispersed graphs, which is a concept we formalize in terms of disk neighborhood systems. This approach allows us to develop fast algorithms for road networks without making any additional assumptions about the distribution of edge weights. In fact, our algorithms can allow for non-metric weights.

[1]  John H. Reif,et al.  A Dynamic Separator Algorithm , 1993, WADS.

[2]  David Eppstein,et al.  Setting parameters by example , 1999, 40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039).

[3]  Thomas Willhalm,et al.  Combining Speed-Up Techniques for Shortest-Path Computations , 2004, WEA.

[4]  Clifford Stein,et al.  Introduction to Algorithms, 2nd edition. , 2001 .

[5]  R. K. Shyamasundar,et al.  Introduction to algorithms , 1996 .

[6]  Prosenjit Bose,et al.  On the stabbing number of a random Delaunay triangulation , 2007, Comput. Geom..

[7]  Rajeev Raman,et al.  Recent results on the single-source shortest paths problem , 1997, SIGA.

[8]  S. Vavasis,et al.  Geometric Separators for Finite-Element Meshes , 1998, SIAM J. Sci. Comput..

[9]  Michael T. Goodrich,et al.  Planar Separators and Parallel Polygon Triangulation , 1995, J. Comput. Syst. Sci..

[10]  Gary L. Miller,et al.  Finding Small Simple Cycle Separators for 2-Connected Planar Graphs , 1986, J. Comput. Syst. Sci..

[11]  Nancy M. Amato,et al.  A Randomized Algorithm for Triangulating a Simple Polygon in Linear Time , 2001, Discret. Comput. Geom..

[12]  Andrew V. Goldberg,et al.  Scaling algorithms for the shortest paths problem , 1995, SODA '93.

[13]  Jeannette M. Wing.  An introduction to computer science for non-majors using principles of computation , 2007, SIGCSE.

[14]  Robert E. Tarjan,et al.  A data structure for dynamic trees , 1981, STOC '81.

[15]  Ulrich Meyer,et al.  Single-source shortest-paths on arbitrary directed graphs in linear average-case time , 2001, SODA '01.

[16]  Andrew V. Goldberg,et al.  Computing the shortest path: A search meets graph theory , 2005, SODA '05.

[17]  Robert E. Tarjan,et al.  Fibonacci heaps and their uses in improved network optimization algorithms , 1984, JACM.

[18]  Franz Aurenhammer,et al.  Voronoi diagrams—a survey of a fundamental geometric data structure , 1991, CSUR.

[19]  David Eppstein,et al.  A Deterministic Linear Time Algorithm for Geometric Separators and its Applications , 1995, Fundam. Informaticae.

[20]  Peter Sanders,et al.  Highway Hierarchies Hasten Exact Shortest Path Queries , 2005, ESA.

[21]  Bernard Chazelle,et al.  Could Your iPod Be Holding the Greatest Mystery in Modern Science? , 2006 .

[22]  Michael Batty,et al.  Fractal Cities: A Geometry of Form and Function , 1996 .

[23]  Philip N. Klein,et al.  Faster Shortest-Path Algorithms for Planar Graphs , 1997, J. Comput. Syst. Sci..

[24]  Bernard Chazelle.  Triangulating a simple polygon in linear time , 1991, Discret. Comput. Geom..

[25]  Gary L. Miller,et al.  Separators for sphere-packings and nearest neighbor graphs , 1997, JACM.

[26]  Pavol Návrat.  Review of "Algorithm design: foundations, analysis and internet examples" by Michael T. Goodrich and Roberto Tamassia. John Wiley & Sons, Inc. 2001. , 2004, SIGA.

[27]  Anne Condon,et al.  RNA Molecules: Glimpses Through an Algorithmic Lens , 2006, LATIN.

[28]  M. Erwig.  The graph Voronoi diagram with applications , 2000 .

[29]  Shang-Hua Teng,et al.  Disk packings and planar separators , 1996, SCG '96.

[30]  G. A. Klunder,et al.  The shortest path problem on large-scale real-road networks , 2006 .

[31]  F. Benjamin Zhan,et al.  Shortest Path Algorithms: An Evaluation Using Real Road Networks , 1998, Transp. Sci..

[32]  Kurt Mehlhorn,et al.  A Faster Approximation Algorithm for the Steiner Problem in Graphs , 1988, Inf. Process. Lett..

[33]  Michael T. Goodrich,et al.  Algorithm Design: Foundations, Analysis, and Internet Examples , 2001 .

[34]  Mikkel Thorup,et al.  Undirected single-source shortest paths with positive integer weights in linear time , 1999, JACM.

[35]  Bernard Chazelle,et al.  Is the thrill gone? , 2005, CACM.

[36]  Franz Aurenhammer,et al.  Voronoi Diagrams , 2000, Handbook of Computational Geometry.

[37]  R. Tarjan,et al.  A Separator Theorem for Planar Graphs , 1977 .

[38]  Jeffrey Scott Vitter,et al.  Shortest paths in euclidean graphs , 2005, Algorithmica.

[39]  Bojan Mohar,et al.  A polynomial time circle packing algorithm , 1993, Discret. Math..